Remark 2.4 While the setting of gambling we consider is equivalent to those in [1]}, [2]}, the literature uses a different convention.
For the two-horse race setting with odds \(\frac{1}{m}\)  and \(\frac{1}{1-m}\)  and a betting strategy \((b_t)_{t=1}^{\infty }\) , they write the multiplicative wealth as
\(\frac{1}{m} \widetilde{c}_t b_t + \frac{1}{1-m} (1-\widetilde{c}_t)(1-b_t)=1+\lambda _t(m) (\widetilde{c}_t-m),\) 
